Question 1153986
The table lists a readings for three years.

​(a) If the quadratic relationship between the carbon dioxide concentration {{{C}}} and the year {{{t }}}is expressed as {{{C=at^2+bt+c}}}​, where {{{t }}}equals {{{0}}} corresponds to​ 1962, use a system of linear equations to determine the constants​ {{{a}}}, {{{b}}}, and​ {{{c}}}, and give the equation.


Year.
1962
1982
2002
CO2
{{{319}}}
{{{340}}}
{{{368}}}


{{{C=at^2+bt+c}}}

 since {{{t}}} equals {{{0}}} corresponds to​ 1962, and {{{C}}}  is {{{319}}}

{{{319=a*0^2+b*0+c}}}
{{{highlight(c=319)}}}


{{{C=at^2+bt+c}}}................year 1982, {{{C=340}}}, {{{t=20}}}, {{{c=319}}}
{{{340=a*20^2+b*20+319}}}
{{{400a+20b=340-319}}}
{{{400a+20b=21}}}.......solve for {{{b}}}

{{{b=21/20-20a}}}.................eq.1


{{{C=at^2+bt+c}}}................year 2002, {{{C=368}}}, {{{t=40}}}, {{{c=319}}}
{{{368=a*40^2+b*40+319}}}
{{{1600a+40b=368-319}}}
{{{1600a+40b=49}}}
{{{40b=49-1600a}}}
 {{{b= 49/40-40a}}}........eq.2


from eq.1 and eq.2 we have

{{{21/20-20a=49/40-40a}}}

{{{40a-20a=49/40-21/20}}}

{{{20a=7/40}}}

{{{highlight(a=7/800)}}}


go to

{{{b=21/20-20a}}}.................eq.1, plug in{{{ a}}}

{{{b=21/20-20(7/800)}}}

{{{highlight(b=7/8)}}}

your equation is:

{{{highlight(C=(7/800)t^2+(7/8)t+319)}}}



​(b) Predict the year when the amount of carbon dioxide in the atmosphere will double from its 1962 level.

1962 level is {{{319}}}
in {{{t }}}years will  double which is  {{{C=2*319=638}}}

{{{638=(7/800)t^2+(7/8)t+319}}}.........sole for {{{t}}}

{{{(7/800)t^2+(7/8)t+319-638=0}}}

{{{(7/800)t^2+(7/8)t-319=0}}}


using quadratic formula, we get


{{{t = 30 sqrt(303/7) - 50}}}=> exact solution  (only positive root, we will disregard negative solution)

{{{t = 147.38}}}-> approximately

 {{{147.38}}} years from 1962 is: year {{{2109}}} and the amount of carbon dioxide in the atmosphere will double from its 1962 level.



check:

{{{638=(7/800)(147.38)^2+(7/8)(147.38)+319}}}

{{{638=(7/800)(147.38)^2+(7/8)(147.38)+319}}}

{{{638=190.05+128.95+319}}}

{{{638=638}}}